Properties

Degree $2$
Conductor $259920$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 2·7-s − 2·13-s + 6·17-s + 6·23-s + 25-s − 4·29-s − 2·35-s − 10·37-s − 8·41-s − 2·43-s − 2·47-s − 3·49-s − 2·53-s + 14·61-s − 2·65-s + 4·67-s − 12·71-s + 6·73-s − 8·79-s + 2·83-s + 6·85-s − 8·89-s + 4·91-s + 6·97-s + 101-s + 103-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.755·7-s − 0.554·13-s + 1.45·17-s + 1.25·23-s + 1/5·25-s − 0.742·29-s − 0.338·35-s − 1.64·37-s − 1.24·41-s − 0.304·43-s − 0.291·47-s − 3/7·49-s − 0.274·53-s + 1.79·61-s − 0.248·65-s + 0.488·67-s − 1.42·71-s + 0.702·73-s − 0.900·79-s + 0.219·83-s + 0.650·85-s − 0.847·89-s + 0.419·91-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 259920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(259920\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 19^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{259920} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 259920,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.418875083\)
\(L(\frac12)\) \(\approx\) \(1.418875083\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 - T \)
19 \( 1 \)
good7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.92317665094899, −12.28454821426484, −12.07503733357592, −11.43911507288035, −10.96871939052627, −10.33777050690007, −10.02231130026400, −9.654827237564252, −9.226548602201995, −8.576524469242836, −8.293270987762998, −7.473533577132942, −7.179065175592622, −6.691958358770027, −6.227715864620002, −5.530678844385170, −5.196722115925360, −4.867207114539798, −3.879131325563981, −3.511763857703667, −3.002046042125418, −2.503487223742229, −1.667598727255265, −1.247508300860976, −0.3244235800575461, 0.3244235800575461, 1.247508300860976, 1.667598727255265, 2.503487223742229, 3.002046042125418, 3.511763857703667, 3.879131325563981, 4.867207114539798, 5.196722115925360, 5.530678844385170, 6.227715864620002, 6.691958358770027, 7.179065175592622, 7.473533577132942, 8.293270987762998, 8.576524469242836, 9.226548602201995, 9.654827237564252, 10.02231130026400, 10.33777050690007, 10.96871939052627, 11.43911507288035, 12.07503733357592, 12.28454821426484, 12.92317665094899

Graph of the $Z$-function along the critical line